La preservación de los recursos humanos en el sector pesquero

Shay’s Articulation of Uncertainty: From Global to Probabilistic View. Shay demonstrated a global view of the sampling distribution in the first two stages. He was aware and certain of the signal in this distribution around 20%. But it was the noise he noticed in the distribution that caused him to feel uncertain about the ability to make conclusions based on random samples of size 70. Yet, he expressed his un- certainty only in general terms, for example that “it is not accurate enough.” Only in the beginning of the third stage, motivated by his deep curiosity to understand uncer- tainty, he started viewing the sampling distribution probabilistically, which enabled him to quantify the uncertainty.

At the third and fourth stages, Shay interpreted the bin heights as probability state- ments about a sample statistic. At the end of Stage 5, he explained that he looked for the “probability that it [%ROCK in sample size 70] would turn out accurate and ac- curate enough.” But he realized not only “that it is not accurate enough, it is also a probability that is not sufficiently high.” Then he determined that a sample size of 70 was “unequivocal not enough” and increased the sample size to 100 to reduce uncer- tainty by getting smaller variability. For example, he realized that the probability of getting a sample statistic in the range of plus minus two deviations is 86% in sample size 100, larger than 75% in sample size 70. Although he did not remember that the probability of 75% on samples size 70 was calculated for getting a sample statistic in a larger range of plus minus three deviations, Shay was not surprised and even expected to get a larger probability on larger samples. Thus, it seems that he under- stood informally two key ideas regarding sampling distributions: (1) as the sample size gets larger, the variability of the sample means gets smaller; and (2) the bins’ relative frequency represents the probability of the sample statistics (Garfield et al., 2005).

Furthermore, when Shay tried to control uncertainty in the fifth stage, he deter- mined a range of statistic values with an “error” of three deviations from 20%, found the probability of obtaining a sample statistic outside of this range, and named it “risk.” It seems that he thus described informally a measure of variability of sam- pling distribution, similar to the formal standard error of the mean. He also described the likelihood of different values of the sample %ROCK in order to quantify uncer- tainty. In order to control and decrease uncertainty, he described the probabilities that a sample will fall in different ranges around the original %ROCK. Therefore, we can claim that he was able to use his knowledge about sampling distributions to describe: a) the size of the standard error of the mean; and b) the likelihood of different val- ues of the sample mean (Garfield et al., 2005). In some sense, Shay’s “discovery” of how to control uncertainty by relating it to the probability of getting a certain statis- tical result can be viewed as a first step towards understanding the reasoning behind hypothesis testing.

Liron’s Articulation of Uncertainty: From Deterministic to Quasi-Probabilistic View. Liron’s articulations were characterized with a local view of uncertainty in the sampling distribution. He noticed from the beginning that most of the %ROCKs in the sampling distribution were equal to the original %ROCK and that the mean of the sampling distribution was very close to the original %ROCK. Focusing on these sig-

DISCUSSION AND IMPLICATIONS 89

nals, Liron expressed a very high level of confidence most of the time and sometimes even an absolute certainty in samples of size 70. Liron’s consideration of one of two possible conclusions (correct or incorrect) also demonstrates his deterministic view of uncertainty (Ben-Zvi et al., 2012). The shift in his view happened during the fifth and sixth stages: Following his discussions with Shay and observing Shay’s actions and articulations, Liron widened his observations to an interval of results around the value of the original %ROCK. When the students began observing a sampling distribution of samples size 100, Liron referred to relative frequency in sampling distribution but still was focused on the mode. With the interviewer’s mediation, he expressed a quasi-probabilistic view when he accounted for chance in the sampling distribution. But in the seventh stage, his decisions were based only on the values’ difference from the original %ROCK and there was no reference again to probabilities or to frequencies.

3.6 Discussion and Implications

This chapter focuses on the question: How can students’ articulations of uncertainty emerge while informally exploring sampling distributions using the integrated mod- eling approach? To address this question we analyzed Shay’s and Liron’s articula- tions of uncertainty in seven stages in which they explored sampling distributions in the model world in order to find the minimal sample size on which they could make ISIs in the data world. They struggled with the fundamental concept of sta- tistical uncertainty in the process of making a statistical inference from a sample to population.

The study sheds light on how young students were able to engage with the com- plex idea of sampling distribution by encouraging them to articulate their uncertainty in the context of making ISIs. Even students with statistical knowledge about theoret- ical probability distributions find it difficult to make connections between theoretical models and empirical distributions (Noll & Shaughnessy, 2012). Both of the students understood that the exploration of the sampling distribution can help them decide on the minimal sample size needed to draw reliable conclusions about the population. Actually, their argumentation circled around the question of whether the sample size explored was large enough or not. That is, they began to connect between repeated samples that were drawn from a theoretical model and a single empirical sample that they were about to collect. This finding strengthens the argument that one needs to envision a process of repeated sampling to understand the logic behind ISI and the relationship between sample and population (Shaughnessy, 2007; Thompson et al., 2007).

We suggest that there was another important factor that helped the students to connect between repeated sampling and a single sample: the students’ engagement with an authentic context (Edelson & Reiser, 2006) and in the data and model worlds. They explored sampling distributions that stemmed from an authentic and real mo- tivation to study students’ music preferences in their age group. The exploration of the sampling distribution came after they realized that they could not ask everyone,